The mathematician Javier Gómez Serrano, born in Madrid 39 years ago, has partnered with the artificial intelligence giant Google DeepMind to try to “soon” solve one of the most devilish enigmas known to humans, the Navier-Stokes equations, as he himself has told EL PAÍS. This is one of the seven Millennium Prize Problems, for whose solution the U.S.-based Clay Mathematics Institute is offering $1 million (and “immortal fame,” as the Spanish researcher emphasizes). The so-called Navier-Stokes Operation, underway for three years with a team of 20 people, has so far been carried out with complete discretion, although the chief of Google DeepMind, Demis Hassabis, let slip in a January interview that they are “close to solving a Millennium Prize Problem” without mentioning which one. “We’ll see that in the next year or year and a half.”
Gómez Serrano, who teaches at Brown University, is speaking publicly about the quest for the first time. “There is a general consensus in the community right now that the problem will be solved soon, but no one knows who will do it or how,” he explains. The challenge dates back to the 19th century, when two mathematicians, the Frenchman Henri Navier and the Irishman George Gabriel Stokes, independently published, in 1822 and 1845, the equations that describe the motion of fluids such as water and air. Based on the temperature, viscosity and initial velocity of the fluid, the equations calculate its velocity at a later time. Two centuries after their enunciation, it is still unknown whether the solutions always maintain a regularity or whether an explosion could occur, a sudden change in behavior, as if a tsunami were triggered in a calm sea. These equations are essential for predicting phenomena as relevant as the weather, catastrophic floods, the movement of an airplane, or blood flow in a human being.
The enigma seems poised to be solved imminently. Gómez Serrano co-leads a team of five academics who met while working at Princeton University and are now scattered across various U.S. institutions. They are two geophysicists — Taiwanese Ching-Yao Lai and Chinese Yongji Wang, authors of complex models for calculating melting ice in Antarctica — and three mathematicians: Australian-British Tristan Buckmaster (the other group co-leader), Spaniard Gonzalo Cao Labora, and Gómez Serrano himself, who was raised in the working-class Madrid neighborhood of Puente de Vallecas.
Great mathematical minds have stumbled trying to solve this challenge, devoting the best years of their academic lives to it, only to find themselves stuck in a dead end. In 2014, Thomas Hou’s team at the California Institute of Technology did achieve a major breakthrough, thanks to a prior simplification of the problem. Hou’s group did not use the Navier-Stokes equations, but an earlier version proposed in 1752 by the Swiss mathematician Leonhard Euler to describe the motion of ideal, viscosity-free fluids. The researchers produced a simulation of a fluid inside a cylinder, which, under certain initial conditions, appeared to give rise to a “singularity”: the sought-after tsunami in a calm sea. Gómez Serrano’s team used artificial intelligence techniques — machine learning neural networks — to refine the solution and understand where and how the singularity forms. Their results, published three years ago, were interpreted as a sign that a solution to the million-dollar problem was imminent.
The Spanish mathematician believes that only three other groups in the world are seriously competing to solve the enigma: the aforementioned Thomas Hou in California; the tandem formed by Egyptian Tarek Elgindi and Italian Federico Pasqualotto, also in the United States; and the team led by Diego Córdoba, a 53-year-old Madrid native who, more than a decade ago, supervised Gómez Serrano’s PhD dissertation at the Institute of Mathematical Sciences in Madrid, on how waves break in the sea.
“The Navier-Stokes problem is tremendously difficult,” he acknowledges. “People haven’t been successful using traditional mathematics. What sets our strategy apart from everyone else’s so far is the use of artificial intelligence. That’s the advantage we have, and we think it can work. I’m optimistic; progress is very, very rapid,” he notes. In his view, the solution will arrive sometime in the next five years.
Gómez Serrano has just participated in another historic breakthrough with Google DeepMind: AlphaEvolve, a new artificial intelligence system that solves complex mathematical problems with unprecedented efficiency. The Spanish professor and his American colleague Terence Tao — considered the greatest living mathematician — trained the program for four months with 50 puzzles. “In 75% of cases, it matches the best human result. In another 20%, it improves on it. A 95% success rate is, frankly, impressive,” says Gómez Serrano.
“I think a trained human, reading the related literature, programming extensively, and preparing for several months, could perhaps achieve this. But AlphaEvolve did it in a day. That’s really the advantage. It can become a tool that greatly accelerates research. It will change the way we do mathematics,” he argues.
The head of Google DeepMind, the British neuroscientist Demis Hassabis, and his American colleague John Jumper won the Nobel Prize in Chemistry last year for creating AlphaFold2, an artificial intelligence system capable of predicting the intricate structures of all 200 million known proteins. The program accomplishes in minutes what previously required months of work. The revolution of the new AlphaEvolve system is that, unlike AlphaFold2 and the program designed for the Navier-Stokes puzzle, it is not created with machine learning to solve a very specific problem, but rather is an extensive language model, like ChatGPT, that solves problems in a wide variety of branches of mathematics without requiring specialized knowledge.
Demis Hassabis has predicted that so-called artificial general intelligence, software with human-like intelligence and self-learning capabilities, will arrive around 2030. Gómez Serrano is more cautious. “There are people, more daring than me, who predict that, within five or 10 years, artificial intelligence will be at the level of the best mathematicians in history. I don’t know, but I know it’s progressing extremely rapidly,” he reflects. “There are two currents: the optimists and the pessimists, who think of Terminator [the 1984 film in which an artificial intelligence rebels against humans]. I believe we will ask more complicated questions, that we will be able to better understand nature and design better materials and better medicines. I believe it will change the world, and I want to believe it will change it for the better.”
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